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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eldiag | Structured version Visualization version GIF version |
Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 36706. (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
bj-eldiag | ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-diagval2 36711 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴))) | |
2 | 1 | eleq2d 2811 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ 𝐵 ∈ ( I ∩ (𝐴 × 𝐴)))) |
3 | elin 3955 | . . 3 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴))) | |
4 | ancom 459 | . . 3 ⊢ ((𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I )) | |
5 | bj-elid4 36704 | . . . 4 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 ∈ I ↔ (1st ‘𝐵) = (2nd ‘𝐵))) | |
6 | 5 | pm5.32i 573 | . . 3 ⊢ ((𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))) |
7 | 3, 4, 6 | 3bitri 296 | . 2 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))) |
8 | 2, 7 | bitrdi 286 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∩ cin 3938 I cid 5569 × cxp 5670 ‘cfv 6543 1st c1st 7989 2nd c2nd 7990 Idcdiag2 36708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7991 df-2nd 7992 df-bj-diag 36709 |
This theorem is referenced by: (None) |
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